The structure of decomposable indistinguishability operators

Decomposable fuzzy relations are studied. Symmetric fuzzy relations are proved to be generated by a single fuzzy subset. For Archimedean t-norms, decomposable indistinguishability operators generate special kinds of betweenness relations that characterize them. A new way to generate indistinguishability operators coherent with the underlying ordering structure of the real line is given in the sense that this structure is compatible with the betweenness relation generated by the relation is developed.